Rosen 4.5 p342.
1. In how many ways can five elements be selected in order from a set with three elements when repetition is
allowed? Perm with repetition; each of five selections has three choices : 35 = 243.
3. How many strings of six letters are there? Perm with repetition; each of 6 letters has 26 choices : 266.
5. How many ways are there to assign three jobs to five employees if each employee can be assigned more than
one job? Perm with repetition; each of three jobs can be done by 5 choices : 53 = 125.
7. How many ways are there to select three unordered elements from a set with five elements when repetition is
allowed? Comb with repetition; three similar things of five types : C(3+5–1, 3) = C(7, 3) = 35.
11. How many ways are there to select 8 coins from a piggy bank with 100 pennies and 80 nickels?
Comb with repetition; eight similar things of two types : C(8+2–1, 8) = C(9, 8) = 9.
13. How many ways are there to store 3000 copies of a book in three different warehouses?
Comb with repetition; 3000 similar things of three types : C(3000+3–1, 3000) = C(3002, 2) = 4504501.
14. How many nonnegative integer solutions exist for the equation x1 + x2 + x3 + x4 = 17?
Comb with repetition; 17 similar things of four types : C(17+4–1, 17) = C(20, 3) = 1140.
21. How many ways are there to distribute six similar balls into nine different bins?
Comb with repetition; six similar things of nine types : C(6+9–1, 6) = C(14, 6) = 3003.
31. How many different strings can be made using all the letters of ABRACADABRA?
There are 11 letters : five As, two Bs, one C, one D, two Rs; Perm with similar objects :
C(11, 5)*C(6, 2)*C(4, 1)*C(3, 1)*C(2, 2) = 11!/(5!2!1!1!2!) = 83160.
33. How many different strings can be made using some or all of the letters of ORONO?
We count the number strings of length one, two, three, four, or five using a tree diagram:
length 1:
N
O
R
3
length 2:
O
R N
O
R
N
O
7
length 3:
O
R O O
R N
O
R
N O
O N O
13
length 4:
O R O O O R O O R N R N O O N O O O N O
20
length 5:
R O O O R O O R O R N O N O O N O O O N
20
Total is 3 + 7 + 13 + 20 + 20 = 63 [Note: read strings vertically or follow diagonal.]
37. How many ways can one fruit be chosen on each of seven days from 3 mangos, 2 papayas, 2 kiwi?
Perm with similar objects : C(7, 3)*C(4, 2)*C(2, 2) = 7!/(3!2!2!) = 210.
41. How many ways are there to deal hands of seven cards to each of five players from a deck of 52 cards?
Perm with similar objects : C(52, 7)*C(45, 7)*C(38, 7)*C(31, 7)*C(24, 7) = 52!/(7!7!7!7!7!17!).
43. What is the probability that each player has an Ace if the 52 cards are dealt to four players?
We divide the number of ways of dealing the aces to separate players : 4! = 24 times the ways of dealing
the remaining cards : 48!/(12!12!12!12!) by the ways of dealing all the cards : 52!/(13!13!13!13!) which
equals 24*134/(52*51*50*49) = 133/(17*25*49) = 2197/20825 . .1055